Solve for $x$ : $ 8|x - 7| - 4 = -6|x - 7| + 10 $
Answer: Add $ {6|x - 7|} $ to both sides: $ \begin{eqnarray} 8|x - 7| - 4 &=& -6|x - 7| + 10 \\ \\ { + 6|x - 7|} && { + 6|x - 7|} \\ \\ 14|x - 7| - 4 &=& 10 \end{eqnarray} $ Add ${4}$ to both sides: $ \begin{eqnarray} 14|x - 7| - 4 &=& 10 \\ \\ { + 4} &=& { + 4} \\ \\ 14|x - 7| &=& 14 \end{eqnarray} $ Divide both sides by ${14}$ $ \dfrac{14|x - 7|} {{14}} = \dfrac{14} {{14}} $ Simplify: $ |x - 7| = 1$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 7 = -1 $ or $ x - 7 = 1 $ Solve for the solution where $x - 7$ is negative: $ x - 7 = -1 $ Add ${7}$ to both sides: $ \begin{eqnarray} x - 7 &=& -1 \\ \\ {+ 7} && {+ 7} \\ \\ x &=& -1 + 7 \end{eqnarray} $ $ x = 6 $ Then calculate the solution where $x - 7$ is positive: $ x - 7 = 1 $ Add ${7}$ to both sides: $ \begin{eqnarray} x - 7 &=& 1 \\ \\ {+ 7} && {+ 7} \\ \\ x &=& 1 + 7 \end{eqnarray} $ $ x = 8 $ Thus, the correct answer is $x = 6 $ or $x = 8 $.